New BIRS Scientific Director Appointed 5, 11, 17, 23, 29 Alejandro

16 Pages·2004·10.79 MB·English

PIMS Magazine Vol. 8 Issue 1 Fall 2004
Alejandro Adem PIMS Deputy Director The scenic photographs in this magazine showcase some of the
exceptionally beautiful surroundings of the various PIMS sites
m
e PIMS is pleased to announce the appointment of Alejandro Adem
d
A as PIMS Deputy Director effective January 1, 2005. Until then
o
r
d Manfred Trummer will continue as Deputy Director.
n
a
ej Professor Adem is a very distinguished topologist who has made
Al
important contributions to the study of group actions and to the
cohomology of finite groups. He received his Ph.D. from
Princeton University in 1986. He was the Szegö Assistant Pro-
fessor at Stanford University (1986–89) and a member of the Institute for Advanced
Study (1989–90) before moving to the University of Wisconsin-Madison.
He has held visiting positions at MSRI (1990), ETH Zurich (1993–94), Max Planck
Institut Bonn (1997–98), University of Paris VII (1998), CRM Barcelona (1998), Univer-
sity of Paris XIII (2000), and Hong Kong University of Science and Technology (2001).
See New Deputy Director on page 3
VIGRE Award to the University of Washington
Contributed by Gunther Uhlmann, PIMS-UW Site Director
The University of Washington (UW) has been awarded a five-year grant totaling
US$3.9M by the National Science Foundation (NSF) under the VIGRE program
of the NSF Division of Mathematical Sciences.
See VIGRE on page 4
Inside this Issue 5, 11, 17, 23, 29
Long Arithmetic
Director’s Notes 2
Progressions of Primes
Call for Scientific Proposals 2
Ben Green on page 10 WWrriittiinngg--oonn--SSttoonnee PPrroovviinncciiaall PPaarrkk,, nneeaarr MMiillkk RRiivveerr,, SSoouutthheerrnn AAllbbeerrttaa
Personnel Changes at PIMS 3 PPhhoottoo CCoouurrtteessyy ooff TTrraavveell AAllbbeerrttaa
PIMS Scientists Receive Honours 3
New BIRS Scientific Director Appointed
PIMS PDFs for 2004 4
Banquet in Honour of Barry McBride 4 The Board of Directors of the Pacific Institute for the Mathematical Sciences (PIMS), and the
Citation for Nassif Ghoussoub 5 Board of Trustees, Mathematical Science Research Institute (MSRI), are happy to announce
Call for Proposals for BIRS 2006 5 the appointment of Dr Nassif Ghoussoub as Scientific Director of the Banff International Re-
BIRS 2005 Calendar 6 search Station (BIRS) for a period of 3 years starting January 1, 2004. Dr Ghoussoub replaces
Dr Robert Moody who has led BIRS through its difficult initial stages.
Creative Scientific Writing at BIRS 7
Dr Ghoussoub is presently Professor of Mathematics and “Distinguished
Collaborative Research Groups 9
bu University Scholar” at the University of British Columbia. He is the found-
Arithmetic Progressions of Primes 10 o
ss ing Director of PIMS (1996–2003), a co-founder of MITACS (Network of
u
PIMS Industrial Programmes 11–12 o
h Centres of Excellence in Mathematics of Information Technology & Com-
G
MITACS News 13 s fi plex Systems) and a member of its Board of Directors for the period 1998–
s
NPCDS News Page 13 Na 2003. He is also a co-founder of BIRS and was Chair of its executive
PIMS Education Programmes 14–15 committee for the period 2000–2003.
See New BIRS Scientific Director on page 5
Director’s Notes its prestige and the fact that BIRS is attracting VIGRE grant (see this newsletter), and closer
many excellent mathematical scientists to the cooperation with PIMS was a key element in
Canadian West. Again, we must work hard to the grant application.
build up and maintain the reputation of BIRS. PIMS had another very busy summer with
This will be done by being uncompromising on numerous conferences, workshops and summer
quality, by being proactive in seeking out prom- schools. Special thanks go to the organisers of
ising developments of mathematics, and by im- these events, and to the staff at all PIMS sites
proving the research conditions at the Banff for their invaluable support. I would also like
Center (for instance, by developing on-line li- to mention the many educational events that
brary resources). We are fortunate to have have taken place during the year. PIMS per-
Nassif Ghoussoub as a Scientific Director of sonnel have put an extraordinary amount of
BIRS, who will impart his energy and vision to enthusiasm and energy into organising these
that task. events, and they are indeed an important com-
It is also very good news for us that federal ponent of our mission. I am also pleased to
funding for MITACS has been renewed for a announce that PIMS is making a special effort
second seven-year period. This will certainly towards the First Nations, and that we will be
Ivar Ekeland, PIMS Director
be of help to PIMS in our own efforts to de- represented at the upcoming Provincial Con-
velop industrial projects. We all know how im- ference on Aboriginal Education.
portant it is, for our research and for our stu- Since June, scientists at PIMS-UBC have
What do we have to show for the 2003-04 aca-
dents, to have good industrial contacts, and how taken advantage of the newly added space at
demic year? First and foremost, the science,
difficult and time-consuming it is to establish our Central Office. I wish to thank UBC for
of course. Since April 2004, we have seven
and maintain them. PIMS has succeeded in doubling the space available to PIMS. For those
PIMS Collaborative Research Groups simulta-
doing so, as the GIMMC and the IPSW have of you who know the PIMS site at UBC, we
neously working in periods of concentration, and
demonstrated every year, but at a huge cost in now have expanded into the first floor, and we
26 post-doctoral fellows at the PIMS universi-
time and effort. We are confident that MITACS have an interior staircase joining the first and
ties. This has been generating a lot of scien-
will help us overcome that barrier, and assist second level, in the best mathematical tradition.
tific activity at the PIMS sites, and it has also
us in exploring new directions. The appointment We also have a new lecture/meeting room, and
been attracting a lot of attention. Several promi-
of Gary Margrave as PIMS Industrial Facilita- a small library. During the building period, there
nent research institutes in mathematics around
tor fits into that strategy. was some disruption, as you may easily guess
the world are inquiring about the possibility of
Other good news comes from the South. and as the pictures illustrate.
associating themselves with PIMS.
The University of Washington, which has I hope you enjoy this newsletter. Many
While Ben Green was a post-doctoral fel-
worked with us informally but closely for several thanks to Heather, Stefanie and Manfred for
low at PIMS, he and Terry Tao have obtained
years, has recently been awarded a prestigious putting it together.
a remarkable result on prime numbers (see
Green’s article in this newsletter). This is a tes-
timony to the quality of the research at PIMS,
and the excellent post-doctoral fellows we are
getting.
We must continue to work hard to keep the
momentum, and put every effort into quality to
make PIMS a leading research institute in the
world. To do this, we will be helped by the suc-
cess of the Banff International Research Sta-
tion, which is now almost through its second
year of operations. BIRS is a huge success,
and all the PIMS universities are benefiting from PIMS-UBC Site Renovations: The new lecture room (left), and the interior staircase (right).
Call for Scientific Proposals in the Mathematical Sciences
PIMS now welcomes applications for support for conferences, workshops, seminars and related activities in the Mathematical Sciences, to
occur after April 1, 2005. PIMS also invites scientists of PIMS universities to submit letters of intent for a period of concentration of a
PIMS Collaborative Research Group.
The deadline for applications is October 15, 2004. The results will be announced by January 31, 2005.
For more information please see: http://www.pims.math.ca/opportunities/proposals.html.
2 Vol. 8, Issue 1
Personnel Changes at PIMS
PIMS Scientists Receive
Honours
In addition to the appointment of Alejandro Gerald Cliff is the new
Adem as Deputy Director, PIMS is pleased to PIMS Site Director at the The Canadian Mathematical Society (CMS)
announce the following: University of Alberta. His has selected Joel Feldman (UBC) as the
term began August 2004. winner of the 2004 Jeffery-Williams Prize
New Board of Directors
Prior to that Bryant Moodie and Jingyi Chen (UBC) as the winner of
The five founding universities have appointed rep- was Acting Site Director. the 2003 Coxeter-James Prize.
resentatives as follows: Manfred Trummer The Jeffery-Williams Prize recognizes
(cid:127) Bruce P. Clayman, VP Research, SFU Gerald Cliff and Gunther Uhlmann mathematicians who have made outstand-
(cid:127) Gary Kachanoski, VP Research, U of A continue as Site Directors ing contributions to mathematical research.
(cid:127) Donald E. Brooks, Associate VP Research, UBC at SFU and UW, respectively. Joel Feldman gave the 2004 Jeffery-Will-
(cid:127) Michael Boorman, Dean of Science, U of C Please see http://www.pims.math.ca/ iams Prize Lecture at the CMS Summer
(cid:127) Martin Taylor, VP Research, UVic whatsnew/new_site_directors.html. Meeting, hosted by Dalhousie University,
In accordance with the member universities’ in June 2004.
PIMS Industrial
agreement about PIMS, the university represen- The Coxeter-James Prize recognizes
Coordinator
tatives, together with the PIMS director, have young mathematicians who have made out-
invited three mathematical scientists from the Gary Margrave is now the standing contributions to mathematical re-
PIMS community to join the renewed Board of PIMS Industrial Coordina- search. Jingyi Chen presented the 2003
Directors: tor. This is a new position Coxeter-James Prize Lecture at the CMS
(cid:127) Mark Lewis, Math and Stat Sciences, U of A designed to expand PIMS Winter Meeting, hosted by SFU, in Decem-
(cid:127) Edwin A. Perkins, Mathematics, UBC activities in the industrial ber 2003.
(cid:127) Hugh C. Williams, Math and Stat, U of C sector. Gary Margrave The two joint winners of the CRM-Fields
The following four people have been appointed prize for mathematics for 2002–2003 were
PIMS Education Coordinator in BC
from the industrial sector: John McKay (Concordia) and Edwin
Melania Alvarez-Adem will be the new Edu-
(cid:127) Hugh Morris, President and CEO, Padre Resource Perkins (UBC). The prize, awarded annu-
cation Coordinator in BC, starting January 1,
Management, Delta (former Chair of the PIMS ally by the Centre de Recherches
2005. Her role will complement that of Klaus
Board) Mathématiques in Montreal and the Fields
Hoechsmann, PIMS Education Facilitator, giv-
(cid:127) Brian Russell, VP, Veritas DGC Inc, Calgary Institute in Toronto recognises exceptional
ing a boost to PIMS education activities in BC.
(cid:127) Don Denney, Manager of Process Automation, contributions by a mathematician working
Syncrude Canada Ltd, Edmonton Pi in the Sky in Canada. Ed Perkins is a member of the
(cid:127) Haig Farris, President, Fractal Capital Corp. Managing Editor PIMS Board of Directors.
Michael Boorman is the new Chair of the Nassif Ghoussoub received an Honorary
David Leeming is now the
PIMS Board of Directors. Doctorate from the University of Paris IX in
Managing Editor of Pi in the
To read the biographies of the board mem- June of this year. Nassif is a former director
Sky magazine. For the full
bers please go to http://www.pims.math.ca/ of PIMS and he is now the Scientific Direc-
list of Editorial Board Mem-
personnel/directors.html. tor of BIRS. The citation for the honorary
bers see page 14. David is
degree is on page 5 of this newsletter.
New Site Directors David Leeming also active in various PIMS
Gemai Chen became the new PIMS Site Direc- education initiatives.
tor at the University of Calgary on June 1, 2004. For the full announcement about the three new po- New Deputy Director
He took over from Gary Margrave. sitions listed above please see http://
Continued from page 1
Christopher Bose began his term as Univer- www.pims.math.ca/whatsnew/new_others.html.
Professor Adem's awards include the A.P.
sity of Victoria PIMS Site Director on July 1,
Administrative Sloan Doctoral Dissertation Fellowship (1985),
2004. He took over from David Leeming.
Changes the National Science Foundation Young Inves-
tigator Award (1992), the Romnes Faculty Fel-
On February 1, 2004 Shelley
lowship (1995), and the Vilas Associate Award
Alvarado was appointed
(2003).
Chief Operations Officer.
Professor Adem brings to PIMS high scientific
Andrea Hook has left PIMS
prestige and considerable administrative experience.
to pursue a career in teach-
He has been Chair of the Department of Mathemat-
ing. Fanny Lui is the new
ics at the University of Wisconsin-Madison (1999–
PIMS Administrator.
2002), and he is currently co-chair of the Scientific
Andrea Hook at her
Gemai Chen Christopher Bose farewell reception Advisory Committee of MSRI.
Fall 2004 3
A Banquet in Honour of Barry McBride
PIMS PDFs for 2004
On Friday May 21, 2004, PIMS and MITACS held a banquet in honour of Barry McBride at
PIMS is pleased to announce the PIMS Postdoctoral Fel-
the University Golf Club near UBC. The banquet celebrated one of many legacies of Barry
lows (PDFs) for 2004. The members of the review panel
McBride: His defining role in the remarkable development of the mathematical sciences in
were Kieka Mynhardt (UVic), Chen Greif (UBC), Derek
western Canada during the last decade.
Bingham (SFU), Sudarshan Kumar Sehgal (UA) and
Barry McBride recently finished his term as Vice-President Academic and Provost of
Michael Lamoureux (UC), chair of the review panel.
the University of British Columbia completing several decades of dedicated service to
New PIMS PDFs: UBC and to Canada, both as an inspirational leader and as a tireless advocate for Canada's
Omer Angel: Probability. Supervised by Gordon quest for academic excellence.
Slade (UBC). Notable among Barry's many achievements are his crucial contributions to the success
Wael Bahsoun: Measurable Dynamics. Chris Bose of the mathematical science community in BC and Alberta with its hugely positive impact in
(UVic). the rest of Canada and the Pacific Northwest. Barry's leadership and unwavering support
Boaz Ben-Moshe: Computational Geometry. Binay were instrumental in the creation of the Pacific Institute for the Mathematical Sciences
K. Bhattacharya (SFU). (PIMS), the MITACS Network of Centres of Excellence, the Banff International Research
Shlomo Hoory: Expander Graphs. Joel Friedman Station (BIRS), as well as in the emergence of UBC’s mathematical science community to
(UBC). a position of worldwide prominence.
Hosne Ara Jasmine: Computational Fluid Mechan-
ics. Bruce R. Sutherland (UA).
Antonia Kolokolova: Computational Logic. Eugenia
Ternovska (SFU).
Youngsuk Lee: Computing and Atmospheric Model-
ling. Mary Catherine Kropinski/David J. Muraki (SFU).
Wilson Lu: Statistics. Derek Bingham (SFU).
Mario Pineda-Krch: Mathematical Ecology. Michael
Doebeli (UBC).
Jens Rademacher: Dynamics/Scientific Computing.
Michael Ward/Ralf Wittenberg (UBC/SFU).
Jonathan Walgate: Quantum Information Science.
Barry C. Sanders (UC).
2003 PIMS PDFs Renewed in 2004:
Lyonell Boulton: Spectral Theory of Linear Opera-
tors. Paul Binding (UC).
Dominic Brecher: String Theory CRG (UBC).
Jae-Hun Jung: Theoretical Chemistry, Spectral Meth-
ods. Bernie Shizgal (UBC).
Kyungkeun Kang: Nonlinear PDEs. Stephen Gustafson/ Dinner speeches celebrating the contributions of Barry McBride
Top row: John Waterhouse, VP Academic at SFU; John Hepburn, Dean of Science, UBC with
Tai-Peng Tsai (UBC).
Nassif Ghoussoub; Indira Samarasekera, VP Research at UBC.
Friederich Littman: Number Theory CRG (UBC/SFU).
Second row: Arvind Gupta, Scientific Director of MITACS, and Ivar Ekeland, Director of PIMS;
Frithjof Lutscher: Mathematical Ecology and Barry McBride; Nassif Ghoussoub, BIRS Scientific Director, MC for the event.
Evolution CRG (UA/UC). Last row: Ivar Ekeland and Barry McBride; Barry McBride, John Waterhouse and Ken Foxcroft
(former Board Member of PIMS).
Jan Manuch: Computer Science, Communication Com-
plexity. Arvind Gupta (SFU).
Bahram Rangipour: Algebraic Topology. John VIGRE grant to UW The principal investigators on the VIGRE grant
Phillips (UVic). Continued from page 1 are Douglas Lind, Loyce Adams, Peter
Anne-Gaelle Rolland-Lagan: Computer Science, Mod- The main goal of VIGRE is to attract and Guttorp, Selim Tuncel, and Kai-Kat Tung. One
elling and Simulation in Mathematics. Przemyslaw retain more US citizens and permanent resi- of the components of this VIGRE award is
Prusinkiewicz (UC). dents into the mathematical sciences at all integration with PIMS in order to create new
Chris Rowe: Number Theory CRG (UBC/SFU). levels. This award builds on the accom- research opportunities for students, postdocs,
Jacob Shapiro: Algebraic Geometry, Gromov-Witten plishments of a previous VIGRE award to and faculty of the University of Washington.
Invariants. Kai Behrend/Jim Bryan (UBC). the UW in 1999, one of the first given, but The VIGRE award will further strengthen
Vardarajan Suneeta: Don Page (UA). at a level that is nearly double that of the the ties between PIMS and the University of
Jian-Jun Xu: Scientific Computing CRG (SFU). previous award. The competition was keen: Washington, and provide new avenues to sup-
Jianying Zhang: Scientific Computing CRG (UBC). out of 29 proposals only three were funded. port our collaborations.
4 Vol. 8, Issue 1
Citation for Dr Ghoussoub’s
Call for proposals for
Honorary Doctorate
BIRS 2006
The Banff International Research Station for
Mathematical Innovation and Discovery (BIRS)
is now accepting proposals for its 2006
programme.
Complete information, guidelines, and online
forms are available at the BIRS website http://
pims.math.ca/birs.
BIRS will be again operating for 40 weeks
in 2006 and will be running each week, either a
full workshop (40 people for 5 days) or two
half-workshops (20 people for 5 days). BIRS
provides full meals, accommodation, and re-
search facilities at no cost to organisers and
invited participants, in a setting conducive to
research and collaboration.
The deadline for 5-day workshop proposals
and Summer Schools is October 15, 2004.
In addition, BIRS will host 2-day workshops.
BIRS will also operate its Research in Teams
and Focused Research Groups programmes,
which allow smaller groups of researchers to get
together for several weeks of uninterrupted work
at the station.
October 15, 2004 is also the preferred date to
apply for these programmes. Proposals for projects
other than 5-day workshops, however, can be sub-
mitted anytime — subject to availability — at least
4 months before their requested start date.
If possible, proposal submissions should be
made using the online submission form at http://
www.pims.math.ca/birs/.
Citation by Ivar Ekeland
New BIRS Scientific Director
Continued from page 1
Dr Ghoussoub was the recipient of the Coxeter-James prize in 1990, of a
Killam senior fellowship in 1992 and has been a fellow of the Royal Society
of Canada since 1993. He was chair of NSERC's grant selection committee
for mathematics in 1995–1996 and vice-president of the Canadian Math-
ematical Society from 1994 to 1996. He was co-editor of the Canadian Jour-
nal of Mathematics from 1993 to 2002 and is currently on the editorial board
of various international journals. His present research interests are in non-
linear analysis and partial differential equations.
As Scientific Director of BIRS, Dr Ghoussoub will be responsible for the
overall functioning of the scientific and intellectual activities of the Station,
and will act as an ambassador and as its public representative. He will chair
its Scientific Advisory Board and its Steering Committee and will make sure
that all scientific activities of the Station are run at the standards and with the
The Pre-Arrival BIRS Team: Jacqueline Kler (BIRS Programme Coordinator),
Dil Bains (UVic Admin Assistant) and Barbara Dempsey (BIRS Admin Assistant) integrity expected by its sponsoring foundations and granting councils.
Fall 2004 5
Banff International Research Station: 5-Day Workshops in 2005
Mar 12–17 Dynamics, Probability, and Conformal (Toronto), E. Meinrenken (Toronto), Y. Karshon Sept 10–15 Interactions Between
Invariance: P. Jones (Yale), M. Yampolsky (Toronto), (Toronto), T. Holm (Berkeley) Noncommutative Algebra and Algebraic
I. Binder (Toronto), S. Rohde (Washington) Geometry: C. Ingalls (New Brunswick), M. Artin
May 28–June 2 Critical Scaling for Polymers
(MIT), Z. Reichstein (UBC), L. Small (UC, San Diego),
Mar 19–24 Computational Fuel Cell and Percolation: D. Brydges (UBC), J. Chayes
J. Zhang (Washington)
Dynamics-III: B. Wetton (UBC), K. Promislow (Microsoft Research), G. Slade (UBC)
(Michigan State), J. St. Pierre (Ballard) Spet 17–22 Order, Disorder, and Transport:
June 4–9 Mathematical Issues in Molecular
Recent Advances in Schrödinger Operator
Mar 26–31 Representations of Kac-Moody Dynamics: P. Tupper (McGill), R. Skeel (Illinois UC)
Theory: P. Hislop (Kentucky), R. Froese (UBC),
Algebras and Combinatorics: V. Chari
June 11–16 Geometric and Asymptotic Methods V. Jaksic (McGill), A. Klein (UC, Irvine)
(UC Riverside), G. Cliff (Alberta), P. Littelmann
in Group Theory: A. Rhemtulla (Alberta), M. Sapir
(Wuppertal), N. Reshetikhin (Berkeley) Sept 24–29 Time-frequency Analysis and
(Vanderbilt), R. Grigorchuk (Texas A&M),
Nonstationary Filtering: G. Margrave (Calgary),
Apr 2–7 Workshop in Homotopical A. Olshanskiy (Vanderbilt), D. Wise (McGill)
H. Feichtinger (Vienna), K. Groechenig (Connecticut),
Localization and the Calculus of Functors:
June 18–23 Combinatorial Game Theory M. Lamoureux (Calgary)
G. Peschke (Alberta), K. Bauer (Calgary),
Workshop: R. Nowakowski (Dalhousie),
R. Cohen (Stanford), H. Sadofsky (Oregon) Oct 1–6 Challenges in Linear and Polynomial
E. Berlekamp (Berkeley), M. Müller (Alberta), D. Wolf
Algebra in Symbolic Computation Software:
Apr 9–14 Complex Data Structures: (Gustavus Adolphus)
E. Kaltofen (North Carolina State), W. Decker
J. Stafford (Toronto), J. Berger (SAMSI),
July 9–14 Rigidity, Dynamics, and Group (Saarlandes), K. Geddes (Waterloo), S. Watt
M. Thompson (Waterloo), N. Reid (Toronto)
Actions: D. Fisher (Lehman, CUNY), R. Spatzier
(Western Ontario)
Apr 16–21 Numerical Relativity: R. LeVeque (Michigan), D. Witte Morris (Lethbridge)
Oct 8–13 Progress in Algebraic Geometry
(Washington), M. Choptuik (UBC), L. Lehner
July16–21 Mathematical Biology of the Cell:
Inspired by Physics: M. Thaddeus (Columbia),
(Louisiana State), D. Arnold (Minnesota),
Cytoskeleton and Motility: L. Keshet (UBC), J. Bryan (UBC), R. Vakil (Stanford)
E. Tadmor (Maryland)
A. Mogilner (UC Davis), P. Janmey (U Penn),
Oct 15–22 MITACS Week
Apr 23–28 Applications of Torsors to Galois E. Cytrynbaum (UBC)
Cohomology and Lie Theory: A. Pianzola (Alberta), Oct 22–27 Visco-plastic Fluids: from Theory to
July 30–Aug 4 Renaissance Banff: Mathematics,
V. Chernousov (Alberta), S. Kumar (North Carolina), Application: I. Frigaard (UBC), N. Balmforth
Music, Art, Culture: R. Moody (Alberta),
D. Harari (ENS Paris) (UC, Santa Cruz)
N. Ghoussoub (UBC), C. Rousseau (Montreal),
Apr 30–May 05 Micro- and Nano-fluidic Systems R. Sarhangi (Towson), J. Morrow (Banff Centre) Oct 29–3 Probabilistic Combinatorics: Recent
Descriptions: J. Harrison (Alberta), J. Santiago Progress and New Frontiers: B. Sudakov
Aug 20–Aug 25 Mathematical Epidemiology:
(Stanford), K. Jensen (MIT) (Princeton), N. Alon (Tel Aviv), B. Reed (McGill),
P. van den Driessche (Victoria), H. Hethcote (Iowa),
V. Vu (UC, San Diego)
May 7–12 Aggregation and Disaggregation: S. Levin (Princeton)
Characterization and Identification of Nov 5–10 Number Theory Inspired by
Aug 27–Sept 1 Topology: R. Stern (UC Irvine),
Collective Demand: P. A. Chiappori (Chicago), Cryptography: D. Boyd (UBC), C. Pomerance
I. Hambleton (McMaster), M. Hopkins (MIT),
I. Ekeland (UBC) (Dartmouth), I. Shparlinski (Macquarie), H. Williams
M. Kreck (Heidelberg)
(Calgary)
May 14–19 Densest Packings of Spheres:
Sept 3–8 Workshop on Analytic and Algebraic
K. Bezdek (Calgary), H. Cohn (Microsoft Research), Nov 19–24 MSRI Hot Topic
Methods in Complex and CR Geometry:
C. Radin (Texas, Austin)
J. Bland (Toronto), Y.-T. Siu (Harvard), J. Kohn Dec 3–8 PIMS Hot Topic
May 21–26 Moment Maps in Various (Princeton), L. Lempert (Purdue), J. D'Angelo
Dec 10–15 Regulators II: J. Lewis (Alberta), V. Snaith
Geometries: E. Lerman (Illinois UC), L. Jeffrey (Illinois UC)
(Southampton)
Focused Research Groups (FRG), Research in Teams (RIT), Summer Schools (SS) & 2-Day Workshops in 2005
Mar 12–26 Analysis, Computations, and Ex- May 28–Jun 11 Influenza Dynamics: Models and Jul 21–23 Connecting Women in Mathematics
periments on Pinch-off in Liquid Jets (FRG): Data (FRG): D. Earn (McMaster), J. Dushoff Across Canada II (2-day): M. Dubiel (SFU), R. Kuske
R. Miura (New Jersey), H. Huang (York), (Princeton), J. Plotkin (Harvard) (UBC), G. de Vries (Alberta)
D. Papageorgiou (New Jersey), M. Siegel (New Jersey)
Jun 9–11 Meeting of Canadian CS Chairs - Aug 6–20 Computing the Continuous Dis-
Mar 17–19 Second Northwest Functional CACS/AIC (2-day): G. McCalla (Saskatchewan), cretely: Integer-point Enumeration in Poly-
Analysis Symposium (2-day): V. Runde (Alberta) K. Barker (Calgary) hedra (SS): M. Beck (MSRI), S. Robins (Temple)
Apr 30–May 14 Speciality of Malcev Algebras Jun 11–18 Hyperplane Arrangements: Aug 25–Sep 3 Symmetries of Extremal Confor-
(RIT): M. Bremner (Saskatchewan), I. Hentzel (Iowa) Cohomology and Rational Homotopy (FRG): mal Mappings (RIT): E. Schippers (Michigan)
G. Denham (Western Ontario)
May 12–14 The Dark Side of Extra Dimensions Nov 5–12 Hyperbolic Geometry and
(2-day): V. Frolov (Alberta), D. Page (Alberta) Jun 22–Jul 4 2005 Summer IMO Training Camp Quasiconformal Mappings (FRG): P. Bonfert-
(SS): B. Sands (Calgary) Taylor (Wesleyan), M. Bridgeman (Boston), R. Canary
May 14–28 Affinizations of Extended Affine
(Michigan), G. Martin (Auckland), R. Schwartz (Mary-
Lie Algebras (RIT): A. Pianzola (Alberta), B. Allison Jun 25–Jul 9 PIMS Summer School (SS)
land), E. Taylor (Wesleyan)
(Alberta), S. Berman (Saskatoon)
Jul 16–30 Topological Methods for Aperiodic
Date TBA The Kneser-Poulsen Conjecture
May 19–21 Convex and Abstract Polytopes (2- Tilings (FRG): I. Putnam (Victoria), Kellendonk
(2-day) : K. Bezdek (Calgary), R. Connelly (Cornell)
day): E. Schulte (Northeastern), T. Bisztriczky (Calgary), (Lyon), L. Sadun (Texas, Austin)
A. Weiss (York) Date TBA Number Theory in Mirror Symmetry
(RIT): N. Yui (Queens)
6 Vol. 8, Issue 1
The Creative Scientific Writing Workshops at BIRS
An interview with and by the organisers: Marjorie Senechal (Smith College) and Chandler Davis (University of Toronto)
In the spring of 2004, tucked in between work- and mathematical ideas. Our first workshop had Yet except for obvious examples, like those you
shops on mathematical logic and foundations, fifteen participants, all highly accomplished, and cited, creative writing about the content of math-
manifolds and cell complexes, Fourier analysis, the second had twenty, ditto. ematics is extremely rare and creative writing
numerical analysis, probability theory and sto- But aren’t you mixing apples and eggs? about the activity of mathematical creation is
chastic processes, mechanics of particles and Talk about mixing! But, with garlic and salt, even rarer.
systems, game theory, economics, social and apples and eggs make an excellent omelet. We That’s why we organised the workshops: to
behavioural sciences, dynamical systems and assumed from the start—and now we firmly encourage practitioners who engage this content
ergodic theory, and quantum theory, BIRS held believe—that non-mathematicians who write in their work. To give them opportunities to
its second five day experimental workshop on creatively about mathematics and mathematicians, discuss important issues, to learn what others
creative scientific writing. The first took place and mathematicians engaged in creative writing, are doing, to encourage each other, to critique
in September, 2003. Curious colleagues have have a lot to teach and learn from one another. current work, to welcome young writers into
asked us, and continue to ask, what these work- Okay, but isn't it confusing to mix all those liter- the field, to spark collaborations, to forge net-
shop were like and what they accomplished. In ary genres? works and build community.
the self-interview that follows, we address these On the contrary! Creative writing is often Then the creative writing workshops’ goals are
and other questions. sparked by cross-genre insights. For example, the same as any other BIRS workshop!
Why hold workshops on creative scientific writ- in our workshops a poet helped a fiction writer Yes, but as we noted in our report to BIRS after the
ing at BIRS or anywhere else? Mathematics is an find a better way to tell the end of his story. A first workshop, our program is, of necessity, highly
art form, so isn’t mathematical writing creative? mathematician nonfiction writer helped a dra- experimental. In the first workshop we followed
Alas, the population that recognizes the creativ- matist extend the ideas of her play, ideas a film- the standard practice of assigning each participant
ity in a mathematical or scientific paper is smaller maker sitting in on their discussions recast in dog- an hour lecture slot. But that didn’t always give
by many orders of magnitude than the number gerel form. A novelist had insightful comments on people the detailed, line-by-line, feedback some
who remember their Latin. By creative scientific poetry. Of course, it helped a lot that we pressed hoped for. And a few people read work they’d
writing we mean something else: mathematical everyone to circulate his or her work in advance. already published, so feedback was moot. We found
and scientific ideas as subjects for poetry, drama, By the time we arrived in Banff, we’d read it all, we needed to set aside time for other things too.
short stories, novels, nonfiction, comic books, thought about it, and were eager to comment. So, for the second workshop we modified the for-
essays, and film. Why should BIRS take the lead in encouraging this? mat in various ways.
Why would anyone write about science and math- Call it “outreach” if you like, part of the larger How did you organise the time?
ematics in this way? Does anyone do it? effort of mathematicians everywhere in these Well, a typical day went like this:
Mathematics is part of world culture, part of days of dwindling funds to explain who we are 8–9: Breakfast in Corbett Hall, BIRS’s head-
the human spirit. It’s as fit a subject for art, and what we do — and why it matters. Or, if quarters
music, and literature as any other. As for who, you prefer, an effort to engage scientists and 9–10: Reflections: the full group meets to dis-
some mathematicians write poetry, fiction, non- mathematicians in a wider world of discourse. cuss, orally or in writing, issues raised or works
fiction, or drama. And there are non-mathemati- The need to create a body of literature around presented the previous day; further ideas and
cian poets, fiction writers, nonfiction writers and mathematics and science is widely acknowledged inspirations.
dramatists whose work engages mathematicians by mathematicians and non-mathematicians alike. 10–11 and 11–12: Two presentations of works-in-
But is there an audience for cre- progress to the full group, followed by discussion.
ative scientific writing, as you 12–1:30: Lunch
describe it? 1:30–4:30: Time free for writing
The popularity of plays like 4:30–6: Parallel sessions — as many as anyone
Proof and biographies like A wished — on works-in-progress in small groups,
Beautiful Mind and The Man two to ten, for line-by-line comments and editing
Who Loved Only Numbers 6–7:30: Dinner in the Banff Centre dining hall
show that there's a large and 7:30–9: Discussions of general issues, or public
growing public eager to share readings with participants in the Banff Centre's
in the great ideas of mathemat- workshop Writing With Style.
ics and science. The creative Tell us about the evening discussions.
writer’s job is not to coerce The first was called, “What, Why, and For
them to eating these things like Whom?” It covered a lot of ground, from la-
medicine hidden in jam, but to menting math phobia and emphasizing the need
convey these ideas through lit- for better science and mathematics education, to
erature instead of formalism. considering the many forms that outreach can
Participants at the 2004 BIRS Creative Scientific Writing Workshop
Fall 2004 7
take. And we lamented the worm in the apple: two groups for informal public readings. A few mation about publishers and agents. Someone sug-
Proof, A Beautiful Mind and other popular works participants in other BIRS and Banff Centre pro- gested we ask BIRS to link our publications to its
wouldn’t have been so successful had the math- grammes attended too. We hosted an evening of website. The Mathematical Intelligencer already
ematician character been sane. poetry, with eleven readers from both groups. encourages creative writing in mathematics, but we
Audiences always prefer demented geniuses, or They hosted a prose reading evening, with fewer want it to do even more. We hope to hold another
flawed ones. Not only scientists and mathemati- readers of course, but again from both groups. workshop at BIRS in the future, in close associa-
cians. Think of Amadeus, about Mozart, and all And were these readings successful? tion with Banff Centre writing programs, and to
those films about van Gogh. Very. Participants in the two groups met one publish an anthology under their auspices.
It’s true, it’s very difficult to portray intellectual another and some of their conversations contin- I can see it now: a hefty tome, the year's Best
creativity of any kind. But the scientific/mathemati- ued at meals the next day. Another important Creative Writing in Mathematics.
cal nut is tougher to crack — sorry, wrong meta- benefit was the opportunity for writers, in both Yes, the hottest item in the bookstore, its sales
phor. I mean, the mad composer or painter or writer groups, to read their work to and get responses topping the year's best short stories, best es-
can be shown composing or painting or writing from audiences outside their usual orbits. says, best mystery stories, best political fiction,
madly, furiously, but in the end he or she produces So where do we go from here? best non-required reading, best recipes, best sci-
something the public can hear or see or read. While In many directions! Workshop participants plan ence and nature writing, best spiritual writing, best
a mathematician, mad or sane, produces a mysti- to stay in touch, and to keep each other informed sports writing, best travel writing, and best erotica.
fying theorem. But on the other hand, the play of the progress of their work. We will share infor- Hors de doute. Merci.
Copenhagen was a great success and the novel
Einstein's Dreams conveys the scientific creative
RRRRREEEEENNNNNAAAAAIIIIISSSSSSSSSSAAAAANNNNNCCCCCEEEEE BBBBBAAAAANNNNNFFFFFFFFFF
process in a beautiful way. And Arcadia, a funny
and chaotic play whose leitmotif is chaos theory,
is a modern classic. The mathematical formalism is
The Banff Centre, the CMS, and PIMS will host
symbolized in its structure.
a 4-day conference bringing mathematicians and
Using a mathematical structure to talk about math
artists together to encourage and demonstrate the
— that reminds me of a sonnet by Edna St. Vincent
many exciting connections that link the world
Millay, “Euclid alone has looked on Beauty
of mathematics to the visual arts, music, sculp-
bare.” The poem's strict form mirrors deduc-
ture, architecture, poetry, computer graphics,
tive geometry’s austere beauty. “Fortunate they
theatre, and indigenous cultures.
Who, though once only and then but far away,
The conference is the first time the very
Have heard her massive sandal set on stone.”
successful international Bridges Conference
Would you say the sonnet form has mathemati-
(hhhhhttttttttttppppp::::://////////wwwwwwwwwwwwwww.....sssssccccckkkkkaaaaannnnnsssss.....eeeeeddddduuuuu/////~~~~~bbbbbrrrrriiiiidddddgggggeeeeesssss/////) is coming
cal affinities?
to Canada, and to celebrate this, one day will
One of us would, the other wouldn't. But that's
be devoted to the life-long interests and con-
a discussion topic for a future workshop. Back
nections of the late Donald Coxeter to the
to your earlier question: our discussion the last RRRRReeeeennnnnaaaaaiiiiissssssssssaaaaannnnnccccceeeee BBBBBaaaaannnnnffffffffff world of artists and the arts, and to ways of
night was, “Where Do We Go From Here?”
AAAAA MMMMMaaaaattttthhhhheeeeemmmmmaaaaatttttiiiiicccccsssss-----AAAAArrrrrtttttsssss continuing and encouraging his legacy.
The last night? Then tell us first about the public
CCCCCooooonnnnnfffffeeeeerrrrreeeeennnnnccccceeeee aaaaattttt ttttthhhhheeeee The conference will also be of great interest
readings.
BBBBBaaaaannnnnffffffffff CCCCCeeeeennnnntttttrrrrreeeee,,,,, BBBBBaaaaannnnnffffffffff,,,,, AAAAAlllllbbbbbeeeeerrrrrtttttaaaaa to teachers at all levels in mathematics.
Well, as you know, BIRS is located in the world-
••••• General Sessions: to introduce new and inno-
renowned Banff Centre. With studios nestled in
JJJJJuuuuulllllyyyyy 3333311111 tttttooooo AAAAAuuuuuggggguuuuusssssttttt 33333,,,,, 22222000000000055555 vative approaches in mathematical connections
the woods, outstanding mentors, excellent per-
in science and art by inviting international schol-
formance spaces and a fine library, the Banff
ars and artists as invited speakers.
hhhhhttttttttttppppp::::://////////wwwwwwwwwwwwwww.....pppppiiiiimmmmmsssss.....mmmmmaaaaattttthhhhh.....cccccaaaaa/////
Centre nurtures aspiring, mid-career, and estab-
••••• Special Sessions: the conference encourages
lished musicians, painters, photographers, writ- RRRRReeeeennnnnaaaaaiiiiissssssssssaaaaannnnnccccceeeeeBBBBBaaaaannnnnffffffffff/////
participants to contribute to the conference by
ers, and actors. Artists love Banff. And Banff
presenting their own works in the Special Sessions.
loves the artists: the centre's world-class exhibi- RRRRReeeeennnnnaaaaaiiiiissssssssssaaaaannnnnccccceeeeeBBBBBaaaaannnnnffffffffff@@@@@pppppiiiiimmmmmsssss.....mmmmmaaaaattttthhhhh.....cccccaaaaa
••••• Workshop Sessions: providing opportunities
tions, public readings, and performances enhance
for participants to watch and work hands-on with
Banff’s appeal to tourists year-round. The BIRS OOOOOrrrrrgggggaaaaannnnniiiiissssseeeeerrrrrsssss::::: RRRRReeeeezzzzzaaaaa SSSSSaaaaarrrrrhhhhhaaaaannnnngggggiiiii,,,,,
established artists who are crossing the math-
leadership hopes BIRS will interact with the CCCCChhhhhrrrrriiiiissssstttttiiiiiaaaaannnnneeeee RRRRRooooouuuuusssssssssseeeeeaaaaauuuuu,,,,, ematics-arts boundaries.
Centre. So in organising our workshops, we
NNNNNaaaaassssssssssiiiiifffff GGGGGhhhhhooooouuuuussssssssssooooouuuuubbbbb,,,,, RRRRRooooobbbbbeeeeerrrrrttttt MMMMMoooooooooodddddyyyyy ••••• Musical and theatrical performances.
worked closely with Carol Holmes and Edna
••••• Conference Proceedings: available at the
Alford of the Banff Centre’s Writing and Pub-
time of the conference — a resource book and
RRRRReeeeennnnnaaaaaiiiiissssssssssaaaaannnnnccccceeeee BBBBBaaaaannnnnffffffffff LLLLLooooogggggooooo dddddeeeeesssssiiiiigggggnnnnneeeeeddddd bbbbbyyyyy
lishing Program. Their “Writing in Style” work-
guide to further exploration of the talks and
RRRRRooooobbbbbeeeeerrrrrttttt JJJJJ..... KKKKKrrrrraaaaawwwwwccccczzzzzyyyyykkkkk
shop and our second workshop took place the
exhibits of the conference.
same week. On two evenings, we merged the
8 Vol. 8, Issue 1
PIMS Collaborative Research
Group Activities Summer 2004
PIMS CRG on Number Theory PIMS CRG on Mathematical Ecology
and Evolution
Probability and Statistical Mechanics CRG:
Robert Wolpert (Duke, former student of Ed Nelson),
Gerald Folland (Washington), Greg Lawler (Cornell)
and Ed Nelson (Princeton) outside PIMS-UBC. They
were part of “Analysis, Probability, and Logic: A
Conference in Honour of Edward Nelson” which was
held June 17–18, 2004.
Number Theory CRG: PIMS Distinguished Chair
Bjorn Poonen (shown above) from UC, Berkeley
Mathematical Ecology and Evolution CRG: PIMS CRG on Topology & Knot Theory
spoke at the Computational Arithmetic Geometry
Participants of the International Conference
Workshop at PIMS SFU, July 5–9, 2004. In April
Differential Equations and Applications in
2004 Sergei Konyagin (Moscow State University)
Mathematical Biology which was held in Nanaimo,
delivered a PIMS Distinguished Chair lecture
BC, July 18–23, 2004. For more information
series at UBC.
please see http://web.mala.bc.ca/math/conference/.
PIMS CRG on String Theory PIMS CRG on Dynamics & Related
Topics
Topology and Knot Theory CRG: This summer the
main activity of this CRG was the “Knots in
Vancouver” programme. The workshop in Knot
Theory and 3-Manifolds was held at UBC, July 19–
23, 2004, preceded by the MSRI-PIMS Summer
Graduate Programme: Knots and 3-Manifolds at
UBC, July 7–20, 2004.
Call for Proposals:
Dynamics and Related Topics CRG: Participants
of the BIRS Workshop on Aperiodic Order:
Dynamical Systems, Combinatorics, and Letters of Intent —
String Theory CRG: Participants in this year’s Operators, May 29–June 3, 2004. Periods of Concentration
Frontiers of Mathematical Physics Conference
for CRGs
posing putside the PIMS-UBC office.
PIMS CRG on Probability and
Statistical Mechanics PIMS invites interested researchers to
PIMS CRG on Scientific Computing submit letters of intent for periods of con-
centration of a collaborative research
group to start in or after April 2006. Dead-
line: October 15, 2004. Letters of intent
should be 2–4 pages long. On advice of
the PIMS Scientific Review Panel the
PIMS Director will invite successful
groups to submit a full proposal for a pe-
riod of concentration. Full proposals are
usually developed in consultation with the
PIMS Deputy Director and Site Directors.
For more details please consult the “Op-
Scientific Computing CRG: There were two PIMS Probability and Statistical Mechanics CRG:
Distinguished Chairs with the Scientific Computing Participants of the Summer School in Probability portunities” section of the PIMS website
CRG in summer 2004. Otmar Scherzer (left) from the in the recently renovated PIMS conference room at pims.math.ca.
University of Innsbruck visited UBC, and Chris Budd at UBC. The summer school took place from May
(right) from the University of Bath visited SFU. 25–June 25, 2004.
Fall 2004 9
Long Arithmetic Progressions of Primes
By Ben Green, Trinity College, Cambridge, Former PIMS Postdoctoral Fellow
InSpring2004, whileIwasaPIMSpostdocinVancouver, TerryTaoof Unfortunately, this result does not apply to the set of primes since they
UCLAandIpublished anarticleaboutarithmeticprogressions ofprime donotformasetofpositivedensity. Indeed, thefamousPrimeNumber
numbers. Whilstthearticlehasnotyetbeenacceptedforpublicationby Theoremtellsusthatthenumberofprimeslessthan isabout .
N N(cid:2)logN
ajournal,itdoesseemappropriatetogiveaverybriefdescriptionofthe OurmainresultisaversionofSzemere´di’sTheoreminwhichtheset
resultsandmethodsinthisnewsletter. isreplacedbyamoregeneral typeof set. Veryroughlyspeaking, we
N
The sequence of prime numbers has been studied for show that if a set is “pseudorandom” then any set containing a
(cid:0)(cid:0)(cid:1)(cid:0)(cid:2)(cid:0)(cid:3)(cid:0)(cid:1)(cid:1)(cid:1) S (cid:0) N
over2000years. Theappealofprimenumberstomathematiciansisper- positiveproportionof containsarbitrarilylongarithmeticprogressions.
S
haps that they areextremelysimple and natural todefine, but it isoften Theset neednothavepositivedensity(itcanactuallybequitesparse).
S
very difficult or impossible to prove facts about them. As an example, Once this result is proved, one must construct a set which is ap-
S
considerthe250year-oldconjectureofGoldbach,whichsuggeststhatev- propriate for the primes. Simplifying matters somewhat for the sake of
eryevennumbergreaterthan2isasumoftwoprimenumbers. Itiseasy exposition,wetake tobesomethinglikethesetof“almost-primes”,that
S
toverifythisforsmallvalues(e.g. , )and istosaynumbers ,allofwhoseprimefactorsareatleast . The
(cid:1)(cid:0)(cid:1)(cid:3)(cid:3)(cid:3)
(cid:2)(cid:4) (cid:5) (cid:6)(cid:7)(cid:8)(cid:1)(cid:6) (cid:6)(cid:4)(cid:4) (cid:5) (cid:9)(cid:3)(cid:8)(cid:2)(cid:1) n n
tocomeupwithheuristicargumentswhichsuggestthattheconjectureis primesformapositivedensitysubsetof ,but isamuchmoretractable
S S
true. Sofar,however,no-onehasbeenabletosupplyaproof. Thesame object than theprimesthemselves. Indeed by “massaging” alittlebit
S
goesfortheTwinPrimeConjecture,whichstatesthatthereareinfinitely onecanshowusingtechniquesofclassicalanalyticnumbertheory,that
S
manypairsofprimeswhichdifferbytwo,suchas and . ispseudorandom.
(cid:10)(cid:1)(cid:0)(cid:2)(cid:11) (cid:10)(cid:9)(cid:6)(cid:0)(cid:9)(cid:1)(cid:11)
Ourworkconcernsarithmeticprogressionsofprimes. Thesequence Inconstructing theauthorsenjoyedaratherlargesliceofluck. In
S
is an arithmetic progression of five primes, and the se- March2004IwasvisitingAndrewGranvilleinMontre´al,andItoldhim
(cid:2)(cid:0)(cid:6)(cid:6)(cid:0)(cid:6)(cid:3)(cid:0)(cid:0)(cid:1)(cid:0)(cid:0)(cid:7)
quence aboutmyworkwithTerry. Atthatstagewedidnotquiteunderstandhow
tomassagethealmost-primesinawaythatwouldenableustoprovepseu-
dorandomness. Andrewwasabletodirectustosomerecent (asyetun-
(cid:2)(cid:12)(cid:0)(cid:6)(cid:6)(cid:1)(cid:13)(cid:1)(cid:3)(cid:12)(cid:4)(cid:1)(cid:7)(cid:3)(cid:8)(cid:9)(cid:9)(cid:2)(cid:9)(cid:12)(cid:3)(cid:1)(cid:13)(cid:4)(cid:7)(cid:2)(cid:13)(cid:12)(cid:4)k(cid:0) k(cid:5)(cid:4)(cid:0)(cid:6)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)
published)workofDanGoldstonandCemYıldırım. Althoughthework
isanarithmeticprogression of23primes, whichisthelongest currently
ofGoldstonandYildirimconcernsaverydifferentproblem,itturnedout
known (this latter example was discovered on July 24, 2004, by Frind,
thatacertainlemmaintheirpaperconstitutedaverygooddefinitionof
JoblingandUnderwood).
S
togetherwithaproofthat ispseudorandom.
Ourresultisthattherearearbitrarilylongarithmeticprogressionsof
S
Themost original part of our workistheproof ofaversionof Sze-
primes.Wecanactuallyprovethatthereisa -termarithmeticprogression
k mere´di’s Theorem relative to an arbitrary pseudorandom set . The ar-
ofprimes,allofwhichareatmost
S
gument isstronglymotivatedbyabranchofmathematicscalledergodic
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:2)(cid:2)k tshioenosrys,inwcheicghrohuansdbprreoavkeidngvewryoruksoeffuFluirnsttehnebesrtugdiyntohfealraittehm19e7ti0cs.progres-
(cid:0) (cid:0)
Weprove, then, thatanysetconsistingofapositiveproportionof
though this spectacularly large bound (a number with approximately S
containsarbitrarilylongarithmeticprogressions. Subsequentlyweapply
(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:2)(cid:2)k digits) is unlikely to be greeted with much excitement by thistothespecificset discussedabove, whichisrelatedtothealmost-
c(cid:2)o(cid:0)mputationalmathematicians! primes. A number ofSmathematicians have remarked to us that this is
We have not really found a new way of understanding prime num- preciselywhywearesuccessful–historyindicatesthatisolatingproper-
bers. The main advance in our work is a better understanding of arith- tiesoftheprimesthemselvesisaverytrickybusiness, butherewehave
meticprogressions. Afamousresultaboutarithmeticprogressionsisthe beenabletoexhibittheprimesasaspecificinstanceofalargerclassof
1975 theorem of Endre Szemere´di. This states that if is a subset of sets(thepositivedensitysubsetsof ). Forexactlythisreason, itwould
A S
withpositivedensity(say containsonepercentofall beoptimistictoexpectourmethodstoshedanylightontheGoldbachor
N (cid:5) f(cid:6)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)g A
naturalnumbers)then containsarbitrarilylongarithmeticprogressions. TwinPrimeconjecturesinthenearfuture.
A
The paper to which this article refers, The primes contain arbitrarily
long arithmetic progressions by B. Green and T. Tao, is available at
http://www.arxiv.org/abs/math.NT/0404188.
Ben Green (Trinity College, Cambridge)
10 Vol. 8, Issue 1